Topics in Geometry: Quantitative Geometry and Topology


This course is an introduction to quantitative and asymptotic techniques in geometry and topology. Quantitative geometry studies how the size or complexity of a map or space affects its geometry and topology, and how maps and spaces behave at large scales or under various asymptotics. We will introduce some of the ideas and methods of quantitative geometry and apply them to questions in areas like geometric group theory and systolic geometry.

Tentative outline:

Basic information

Further reading

Geometric group theory

Geometric group theory studies the geometry of Cayley graphs of groups and other spaces on which groups act. Given a group \(G\), there are typically many spaces on which \(G\) acts freely, cocompactly, and by isometries. One of the main goals of geometric group theory is to find invariants that describe how the geometry of these spaces depends on \(G\). The Dehn function is one such invariant -- if \(G\) acts on \(X\), then \(G\) and \(X\) have Dehn functions with the same growth rate.

Nilpotent groups and subriemannian geometry

In the last few lectures, we've studied the Heisenberg group and its scaling limit. More generally, the scaling limit of any nilpotent Lie group is a subriemannian manifold. There are still many questions about the geometry of surfaces in subriemannian manifolds and isometries and quasi-isometries between them.

Quantitative homotopy theory

One of my goals with this course was to talk about quantitative homotopy theory and some recent work on bounding the growth of homotopy classes. Unfortunately, we ran out of time, but here are some of the references I was planning to use:

Differentiability and rectifiability

In a different direction, the notes for my previous topics course cover quantitative ways of looking at the differentiability of maps between spaces and the rectifiability of subsets in a space.


Transcribed notes

Notes on quantitative topology (Lectures 1-7), transcribed by the class.

Scanned notes